Plotting Propositions – the Mathematics of Persuasion

Sha la la la la la la
La la la
La di da
La di da

Van Morrison (Brown Eyed Girl)

A lot of people have to write as part of their jobs – grant proposals, progress reports, specifications. And there are endless verbal communications – defending code, disputes over features, justifying organization changes, technology explanations, and so on forever.

Well, the good news is that Hollywood can help!

A lot of people find this all very challenging, and, as part of a typical business/ technical education, have no training in writing and talking persuasively. Sometimes they have been told that laying out the facts in their favor will be enough.

Well it isn’t, as they soon find out. What they need is rules, guidelines, even formulas for persuasive communications.

And that’s why I deserve a bonus

There are such rules, those of traditional rhetoric, which are actually taught to lawyers and communications professionals. But not to engineers or mathematicians.

I’ll talk about rhetoric in a future post but here I want to go beyond rhetoric, because in general it’s not enough. In particular it’s of no use for the most persuasive form of communication: stories.

Phil Ochs once said that a good (protest) song is worth a thousand rallies. Stories can be very effective but where do they come from? Are there rules?

There sure are. If you think about Hollywood’s creations, the first word that comes to mind is “formulaic”. In fact the former script writer Robert McKee wrote a famous book, Story, which lays out many of these formulas, and it became a bible for at least some aspiring Hollywood writers.

Story is not easy reading and of course was never intended to help with professional writing. Can we do better? Yes, that’s why I’m here.

A story

But first I want to tell my own short story. Not so long ago I came up with a theory of stories. The idea is that stories have profiles based on the variation of the protagonist’s fortunes. I was very pleased with myself.

Then a post appeared in Hacker News which described a theory developed by the famous author Kurt Vonnegut for his Master’s thesis (which was rejected!). It used graphs like for polynomials or sine waves to capture what he called the “shape” of a story, based on the rise and fall of the protagonist’s fortunes.

The HN post included a link to a video of Vonnegut explaining his theory with example shapes, like that of Cinderella.

Cinderella = cos(𝜋t) + i sin(𝜋t)

I’d been scooped by Vonnegut, by decades.

I was quite depressed by this discovery. Then two things occurred to me. First, the fact that Vonnegut, a successful author, is advancing the theory gives it credibility. I’ve never written fiction successful or otherwise. Who’d listen if it were just me saying it?

More importantly, I soon discovered that Vonnegut had done nothing with his discovery. I found a video of a talk that Vonnegut had given at least twenty years earlier. It was the same talk.

Finally, it became clear to me that my version of Vonnegut’s theory could be applied to professional writing. I had something to say after all. I felt pretty good in the end and began working on this blog post.

The basic pattern

That’s the story – it’s true – and it also illustrates the most basic formula, what I call the double reversal. (Some philosophers call it the negation of the negation.)

I discovered the pattern (for myself) by thinking about the best known Hollywood formula, namely boy meets girl.

You know how it goes. The protagonist, boy, meets a girl (or girl meets a boy, or a boy meets a boy etc) and they hit it off. They are happy for a while but then because of an argument or a misunderstanding or whatever boy loses girl.

Boys meet girl … and each other?

Boy is very unhappy and tries various schemes to get girl back. Eventually something works and they are reunited, on a firmer basis than before. They end up happier than ever.

I thought up a phrase that summarizes the whole trope, namely

La-di-da; oh shoot; la-di-da.

Meaning at first everything is fine, if not downright good; then suddenly things are bad for whatever reason; then the bad stuff ends and all is well again.

This pattern has almost universal application, including to my little story. At first I’m happy having discovered the idea of story profiles. Then I learned that Vonnegut’s shapes are the same idea and felt bummed out – oh shoot. But then it dawns on me that profiles can be used for professional writing and I’m all la di da.

La di da fund me

What the shapes idea reveals about professional writing is that it is (usually) relentlessly positive. A typical grant proposal emphasizes a wonderful track record, a strong team, and solid ideas for future work. La di da, la di da, la di da di da.

Theoretically (according to the rules of rhetoric) this is persuasive but I’m not so sure any more. It doesn’t make a good story. Good stories need some form of adversity or opposition to be overcome. Who’s interested in a story about an invincible superhero who leaps unopposed from one achievement to the next? For that matter, who believes it?

It’s easy to rework a grant proposal along the lines of the basic trope. First you describe the basic problem and the partial results you and others have had. La-di-da.

Then describe running into a serious problem that the standard methods won’t handle. Oh shoot. Admit that you were stumped at first and even wondered if any solution existed (don’t overdo this part).

Then explain how you in fact solved it and how much improved the situation is (la-di-da) and how much you look forward to doing all kinds of things with your new grant.

The Lambdi-da calculus

Writing out la-di-da gets annoying and tiresome so I invented a simpler notation. La-di-da is simply “+” and oh shoot is “-“. Blanks separate what writers call “acts”. So the basic double reversal pattern is + – +.

The simple version expresses only the sign (good or bad) but not the intensity. We can express the intensity with repetition, so that “++” is la-di-da-di-da and “— is shoot-and-rats.

We’ll use parentheses for nested plots. Suppose girl meets boy, girl loses boy, girl tries three promising things to get boy back, the last of which works. This gives us

+ -(- + – – + – – + +) +

Here each individual unsuccessful attempt to get girl back is – + -, meaning struggling for a solution, temporary optimism as the plan is put in effect, then pessimism as it fails. The third ends with la-di-da success and the whole episode ends la-di-da.

[Oh shoot, a commenter points me to the W-plot idea by Mary Carroll Moore. Basically a W plot is a simple version of the above where boy meets girl, boy loses girl, boy devises scheme to get her back, scheme is a disaster, depths of despair, then somehow gets her back anyway and la-di-da. The lambdi-da formula is ++ -(++ —- +++) ++++ more or less. Oh, la-di-da, this doesn’t scoop the lambdi-da calculus, which is more general. Moore’s made a career out of one formula … maybe I’m on to something!]

Bringing them back for more

Grant applicants and TV series show runners face a common problem, namely follow-up. How do you persuade viewers to come back for the next episode? How do you persuade granting agencies to continue funding?

Hollywood has come up with at least two successful formulas: the happy ending and the cliff hanger.


With the happy ending, the action ends with a strong la-di-da scene, often after intense oh shoot action. The lambdi-da expression, with repetition for intensity, might be as simple as ++ — ++++.

The idea is that the transition — ++++ gives you so much joy and relief that you want more, and will tune in next week. The only risk is that tolerance develops and periodically you have to ramp up the intensity of the oh shoot / la-di-da transition. Eventually this leads to plot inflation and jumping the shark.

An example of a happy ending is the final musical numbers in the film Pitch Perfect. It was so joyous it brought back audiences for Pitch Perfect 2, with mixed results.

The other technique, the cliff hanger, means ending the episode on intense, and usually surprise, adversity. Often this follows a la-di-da session that they naively believe to be the end of the matter.

A typical Lambdi-da expression might be ++ — ++++ —–.

The idea is that the viewers expect from experience that in Hollywood oh shoot is inevitably followed by a more intense la-di-da and they don’t want to miss it.


A recent example of a cliff hanger is the second last episode of season two of the Mandalorian, where tiny adorable Grogu is kidnapped by Moff Gideon’s evil red-eyed Dark Troopers.

The danger with the cliff hanger is that the last negative session may put the audience off. The promise of more la-di-da may not be enough to compensate for more intense oh shoot that precedes it.

Both these techniques have this in common, they promise la-di-da , either as more of the same (happy ending) or the inevitable reversal (cliff hanger).

And I propose …

What about grant applications, bonus requests, promotion applications and the like?

The relentlessly positive approach, e.g. ++ ++ ++ +++ , is just the simplest form of a happy ending pattern. I’ve done well, you can expect more of the same in the future. The same can be said of the more sophisticated double reversal pattern, e.g. +++ — ++++, I suggested.

And what about a cliff hanger? A bit risky. You could end your grant proposal by listing some unsolved problems. But don’t make them sound so hard that the granting authorities, in spite of your past successes, don’t believe you capable of solving them.

Or, for your promotion interview, you could mention serious future challenges you’re well equipped to meet. Again, don’t overdo it.

Tune in next post

Well we’ve certainly done well haven’t we. We started from a simple boy-meets-girl trope and developed a whole calculus of plots. And we’ve shown they can be applied to professional rhetoric, like grant proposals.

In a future post we’ll further extend the la-di-da calculus and take on even more impressive proposals, like venture capital funding pitches. You’ll be up to your ankles in money!

Oh wait, I just realized something potentially catastrophic! I haven’t even tried any of this out! It’s completely untested! It could not work at all and leave you unfunded, unbonused and unpromoted!!

Fortunately I have an idea … but look at the word count. Just enough to tell you the key point, namely that


About Bill Wadge

I am a retired Professor in Computer Science at UVic.
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2 Responses to Plotting Propositions – the Mathematics of Persuasion

  1. Dave says:

    cf “W-plot”

  2. Pingback: Plotting Propositions – the Mathematics of Persuasion – Site Title

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